Four years ago, Robert wrote a Circuit Cellar article exploring analog filters, but he concluded that article with a promise to someday discuss the idea of group delay in filtering. That day has now come, as Robert digs into group delay and why it’s significant.
Long-time readers may remember an article I wrote on analog filters some years ago: “Analog Filters for Dummies” (Circuit Cellar 307, February 2016). In that article, I presented the basics of passive and active filters, and talked about standard filter responses, such as those of Butterworth, Chebyshev and others. I also briefly introduced the so-called Bessel filter. I wrote:
“The last common variant, Bessel filters, is a little more complex to understand. A Bessel filter is not very good both in terms of flatness and attenuation, however it has a key advantage in the time domain: Its so-called group delay is nearly flat. That would bring us a little too far here, but these characteristics preserve the shape of the filtered signals. Let’s keep the subject for another article.”
Now, four years later, isn’t it a good time to dig a little more into that topic? What is this “group delay” about? Why would a flat group delay help? In which designs is it a benefit? In this article I will do my best to give you some information about these topics. As always, I will use only engineer-oriented explanations, with circuits and simulations, rather than mathematics. So, take a seat, and enjoy.
Let’s start with a simple, but often misunderstood, aspect of a specific and simple kind of filter: delay lines. As the name implies, a delay line is nothing more than a circuit that delays electrical signals by a given time. If you want to do the experiment yourself, all you need is a roll of cable. For example, 100m of coaxial cable will provide a delay of about:
assuming the propagation in the cable is 80% the speed of light.
Now, what will be the effect of such a delay on a sine wave injected in the cable, assuming that the cable has no loss? Easy! The signal will be delayed, so the phase of the sine wave will be modified. What will the phase shift be? Because the delay is constant, the phase shift will also be constant, right? Well, wrong. The phase shift will be proportional to the frequency of the sine wave. This is illustrated in Figure 1, and an example will help you to understand it.
Assume that the input signal has a frequency of 250kHz. Its period is 1/250,000Hz = 4µs. Therefore, a delay of 0.4µs means a phase shift of 1/10 of a period, or 360 degrees /10 = 36 degrees. However, if the signal frequency is multiplied by 2 (500kHz), its period is 2µs, and the same 0.4µs delay now means a phase shift of 1/5 of a period or 72 degrees. If you’re not averse to easy math, in Figure 1 you will see that the phase shift in radian, noted as Φ, is 2π times the delay (Δt) times the frequency (f). There is also a minus sign, just because the phase is lagging if the delay is positive.
So, a delay line has a phase shift that varies linearly with the frequency of the input signal. All filters that have the same property are called “linear phase” filters. This means that if you plot their phase shift as a function of the signal frequency, you get a straight line. Or, more precisely, a sawtooth because the phase is rolling back every 360 degrees (2π radian), but each segment of the sawtooth has the same slope. This slope is negative, because the delay is positive, due to the minus sign discussed earlier. Let’s summarize. A linear phase filter, like a delay line, has a sawtooth-shaped phase (shown in purple in Figure 2), and its delay is constant whatever the frequency (in green).
The delay I am talking about here is, in fact, a little different from a simple delay. It’s called “group delay.” Why? Delaying a sine wave just results in a sine wave—with maybe with a phase shift. But now imagine that the input signal is not a plain sine wave, but rather, a short burst of a sine. For example, it could be a Gaussian-shaped amplitude modulation, as shown on the bottom part of Figure 2. What will be the output of the filter with such a signal on its input? Probably, it will have roughly the same shape, but its envelope will be delayed. The so-called “group delay” is simply the time lag between the envelope of input burst and the envelope of the amplitude of the output burst. So, group delay means a propagation delay through a filter, measured on the envelope of the signal.
Let’s summarize again. Linear-phase filters delay the envelope of every signal by the same amount of time, called group delay, no matter what is the carrier frequency of the signal. They are called linear phase because their phase shift varies linearly with the frequency.
NON-PHASE LINEAR FILTERS?
Now imagine that you have a more complex filter—no longer phase linear. What will happen? If you look again at the small equations in Figure 1, you will see that the phase slope is proportional to the group delay. So, if the filter is not phase linear—meaning that its phase slope is not constant—then neither is its group delay. Note: For math-oriented readers, the group delay is then the derivative of the phase shift over frequency (with a -1/2π factor).
So, if a filter is not phase-linear, then its group delay is not flat, meaning it has different values for different signal frequencies. Look again at Figure 2 where the far-right column illustrates this. Here I assumed that a given filter has a steeper phase slope in a small range of frequencies. Therefore, the group delay is increased for signals in that frequency range. The group delay for frequency f2, which is inside this range, is higher than for f1, which is outside. What does that mean? Simply that, if its frequency is f1 or f2, a short burst of a sine wave will not experience the same delay again when considering the envelope of the burst.
Ok, but is this a problem? You bet it is! In real life, signals are usually more complex than a simple amplitude-modulated sine. Imagine that this signal is a short, complex sound—like a drum beat. Such a sound is the sum of plenty of amplitude-modulated frequencies. If the filter doesn’t have a flat group delay, then all these frequency components will not experience the same delay through the filter. The resulting signal will be spread over time, and this may jeopardize the shape of the signal completely. And the drum beat may look like an amplified mouse squeak.
Now you understand that group delays may be important. Let’s see what this group delay is for a typical passive filter. As an example, I designed a 7th-order LC low-pass filter, with a cut-off frequency of 1MHz. To be honest, I simply launched my web browser and clicked on the nice LC-filter online calculator tool available on the www.rf-tools.com website (Figure 3) . As shown, I selected a Chebyshev response, in order to have a steep cut-off. The calculated response of the filter has a 1dB ripple in the bass-band, and more than 30dB of attenuation from 3MHz upward.
Nice, but what about its phase linearity and associated group delay? The online calculator tool easily provides it (Figure 4). The phase shift of the filter is shown in blue, and is clearly far from a straight line—in particular close to 1MHz, which is the cut-off frequency of the filter. The corresponding group delay is plotted in red. It is around 1µs in the pass band of the filter, but jumps up to 4µs for frequencies close to 1MHz. In fact, this example shows a common phenomenon. Filters have more or less a flat group delay far from their cut-off frequency, but may have nasty behavior in their transition region—especially if they have steep transitions.
As this notion of group delay may still be a little fuzzy for you, I made a circuit simulation to show you actual waveforms. For that, I reproduced the same Chebyshev LC filter circuit, using a Spice-based simulator. I used the built-in simulator from my CAD tool suite, Proteus (Labcenter Electronics), but you can run the same example using any Spice simulator, such as the very good and free LTspice simulator from Analog Devices. If you prefer real-life components to simulations, you can also use your soldering iron and oscilloscope—the results should be the same. On the left part of the schematic (Figure 5), I added a burst signal generator. To do that in Spice, I used a voltage multiplier block, fed by a sine source (either 700kHz or 980kHz), and by a rectangular pulse generator with 2µs ramp-up and ramp-down transitions and 1µs pulse duration. This provides a nice way to simulate sine bursts.
Now look at the plots at the top of Figure 5. On the left, there is the input signal (green) and output signal (red), when using a 700kHz carrier frequency. The pulse is delayed by roughly 1µs, and this isn’t surprising, since the calculated group delay was 1µs. The middle column shows what happens with a 980kHz sine. The output pulse is significantly wider than the input, and the delay (measured between the maximums of the envelope) is close to 4µs, as expected. The plot on the far right shows what would happen if the sum of these two signals was sent through the filter. The shape of the output signal is very different than the input, isn’t it?
Intuitively, this example may also help you to understand why such a filter has a larger group delay close to its transition frequency. Look again at the middle plot in Figure 5. At 980kHz, the LC cells of the filter resonate strongly. This resonance needs some time to get energized, so the signal takes some time to appear on the output. But, due to this resonance, energy is still stored in the filter when the input signal is shut off, and the output continues to stay active for a while. That’s why the group delay is larger, and why the output pulse is wider than the input.
As explained, a group delay isn’t actually an electrical delay, it’s more a way to describe the relationship between the input and output signal envelopes. Nevertheless, a flat group delay is a must for plenty of applications. Audio is a classic example, but wideband wireless transmissions also typically require paying attention to this this parameter. So, how could you design a filter with a group delay as flat as possible in the passband? Well, W.E. Thomson calculated the optimal solution for us in 1949, based on work done by a German mathematician, Friedrich Bessel (1784-1846). That’s why these filters are called Bessel-Thomson filters, or more simply: Bessel filters.
Unfortunately, there is no free beer in electronics. A Bessel filter has a maximally flat group delay, but has a very smooth transfer curve. Its frequency cut-off region is not as steep as a Chebyshev filter or even a Butterworth filter. All that said, using the online rf-tools.com calculator, I designed such a Bessel filter with the same cut-off frequency of 1MHz. Compare Figure 6 with Figure 3 and you will clearly see the difference. However, there is—as expected—substantial improvement in the group delay (Figure 7). In the full passband of the filter, this group delay stays between 0.45µs and 0.48µs, and the phase is close to linear.
Do you want to see the behavior of such a Bessel filter in the time domain? I ran the same Spice simulation again, but this time using the LC values of the Bessel filter. The result is shown in Figure 8. Compare the input signals (in green) and output signals (in red) at 700kHz and 980kHz, respectively. If you look closely, you will see that the delays on the signal envelopes are similar. This is clearly visible on the right of the two plots. And if you examine what happens when the sum of these two bursts passes through such a filter (Figure 8, right plots), you will see that the waveforms are nearly not distorted. The shape of the signal is unchanged.
NEGATIVE GROUP DELAY?
Now comes the fun part. You now can see that a group delay isn’t actually an electrical delay. Rather, it is a measure of the time delay between the input and output signal envelopes. Of course, a delay can’t be negative. A filter can’t generate an output signal before knowing what will happen on the input, right? That’s true when talking about actual electrical delays, but not so for group delays. In fact, a group delay can be negative!
At this point, I’m guessing some of you might think I have lost some neurons and may no longer be qualified to write for Circuit Cellar, right? Well, let’s see. I told you that the group delay is positive when the slope of the phase is negative, meaning the phase shift is decreasing when the frequency is increasing. Could this group delay be negative if the phase slope were the other way around? Moreover, could this actually happen with real circuits?
The answer to both questions is affirmative. I found a good and simple example on the Applied Radio Labs website , and reproduced it using my Proteus circuit simulator. Once again, you can use the free LTspice instead if you prefer. As shown in Figure 9, the circuit is simply three components in parallel: L, C and R—connected between the signal and the ground. This configuration makes a notch filter (also called a rejection filter). As shown on the left plot in Figure 9, it attenuates any signal around 100MHz. Now look at the phase response (in red). It is decreasing before and after the rejection region, but is increasing close to 100MHz. Therefore, the slope of the phase shift is positive for such a frequency, and the group delay must be negative. Of course, I ran a time-domain simulation for you, with a 100MHz bursted sine applied on the input. The result is shown on the right plot of Figure 9. Look at the input (green) and output (red) curves. The maximum amplitude of the output is actually before the maximum amplitude of the input! I swear, I’ve not cheated.
Of course, there is no magic here, and the filter isn’t anticipating the behavior of the input. This is just an illusion of a time advance. The illusion would disappear if the signal were not a burst of sine and contained an unpredictable event. Such an event would appear on the output after its application to the input. So why does such a circuit seem to predict the future? In fact, because the circuit is a notch filter, it resonates at 100MHz and strongly attenuates a 100MHz signal. However, this attenuation takes some time to occur because the L and C must be energized. That’s why the output signal is reduced by some tens of nanoseconds after the start of the input signal, before the maximum of the input signal. In any case, the measured group delay is indeed negative!
Group delay may not be the easiest concept to understand, but for many system designs, creating a filter without taking into account its phase behavior may not be a good idea. I hope you grasped the concept. I guess I’ve said this a hundred times, I’ll say it again: The only way to really understand something like this is to work with it a little by yourself. Launch LTspice and reproduce my examples. Then modify the circuit and check the phase and group delay behavior. After that, try switching on your soldering iron and reproduce the same thing using actual circuits. You will learn a lot, and you will likely have lot of fun!
Author’s Note: This is the 76th article of my column, The Darker Side. I have topics in mind for upcoming articles, but perhaps there are subjects you’d like me to explore. Or, maybe you would like to give me some feedback. For example, did you prefer my articles on wireless, on strange behavior of passive circuits, on signal processing or on EMC? Email me at email@example.com and let me know what would make you a happier reader!
 Applied Radio Labs http://www.radio-labs.com
“Group delay and phase delay,” Wikipedia
“What is the difference between phase delay and group delay?”
Group delay, by Christopher J. Struck, CJS Labs
Negative group delay example, by Andor Bariska
What is group delay: Iowa Hills software
Group delay 101, by Merlij Van Veen
PUBLISHED IN CIRCUIT CELLAR MAGAZINE • FEBRUARY 2020 #355 – Get a PDF of the issueSponsor this Article